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If x and y are positive integers such that the product of x [#permalink]
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19 Sep 2010, 06:45
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If x and y are positive integers such that the product of x and y is prime, what is the units’ digit of 7^x + 9^y ? (1) 24 < y < 32 (2) x = 1  I get that just from the question stem itself, you know either x or y must be equal to 1, thus B cannot be the answer. I can't figure out how you can tell the units digit of 9 to the Y just from the statement one. Help? OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/ifxandyar ... 42992.html
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Re: If x and y are positive integers such that the product of x [#permalink]
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19 Sep 2010, 08:02
Atrain13gm wrote: If x and y are positive integers such that the product of x and y is prime, what is the units’ digit of 7^x + 9^y ?
(1) 24 < y < 32 (2) x = 1
 I get that just from the question stem itself, you know either x or y must be equal to 1, thus B cannot be the answer. I can't figure out how you can tell the units digit of 9 to the Y just from the statement one.
Help? Since we know xy is a prime, it implies as you rightly pointed that one of them is 1. (1) This implies y is not 1, hence x must be 1. Also, we know y must be a prime since xy is a prime. So y can only be 29 or 31. Either case, y is odd. What you need to note is that the units digit of 9^{Odd number} is always 9. So combining with x=1, I know the units digit of the number in question must be 7+9 or 6. Hence, sufficient (2) Not sufficient, as I know nothing about y So answer is (a)Further explanationunits digits of 9^x x=1 9 x=2 1 x=3 9 x=4 1 ... The pattern is easy to spot and imagine why it happens. As the last digit is always multiplied by 9, leaving either 1 or 9 itself
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Re: If x and y are positive integers such that the product of x [#permalink]
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19 Sep 2010, 08:06
Hi! I see that noone answer your question yet. I won't discuss about (2) because you already gave the solution.
(1) We know that 24 < y < 32 hence x = 1 ! As a matter fact, if we want a product to be prime we need to multiply 1 by a prime number. As y is much greater than 1, x = 1. Besides, I just said that has to be a prime number. So y = 29 or 31! Now let's see how it goes for the unit digit of th power of 9 :
9^0 = 1 9^1 = 9 9^2 = 81 9^3 = 729 9^4 = ...1
so 9^k if k is odd gives the unit digit as 9.... As y = 29 or 31 (odd numbers), (2) is SUFFICIENT
ANS: A.
Hope it's clear.



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Re: If x and y are positive integers such that the product of x [#permalink]
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19 Sep 2010, 09:21
interestingly i think but for the fact that 2 is a prime and an even prime at that, we are not able to answer based on (2). so we know x=1, so y has to be prime. all primes are odds (but for 2), if not for that, would we not be able to say 7 + 9^odd = ends in 6
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Re: If x and y are positive integers such that the product of x [#permalink]
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19 Sep 2010, 11:30
when xy is some prime number, either x = 1 or y =1. 1. 24<y<32 > x =1 , y = 29 or 31 when y =29> 7^1 + 9 ^ 29 = 7 + any number ending in 9 > unit digit 6 when y = 31 > 7^1 + 9 ^ 31 = 7 + any number ending in 9 > unit digit 6 1) is sufficient 2) not sufficient Ans is A
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Re: If x and y are positive integers such that the product of x [#permalink]
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19 Sep 2010, 11:43
one more thing to add here cyclicity of numbers is always good in GMAT to remember  any number ending with 0,1,5,6 raised to any power will have same unit digit as the number itself, cyclicity is 1 any number ending with 2,3,7,8 will have cyclicity 0f 4 for example 3^(n+1) will have same unit as 3^(n+5) or 3^(n+9) or so on.... where n >= 0 3 > 9 > 27 > 81  243 > 729 > xxx7 > xxx1  any number ending with 4,9 will have cyclicity of 2
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Re: If x and y are positive integers such that the product of x [#permalink]
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19 Sep 2010, 16:01
Atrain13gm wrote: If x and y are positive integers such that the product of x and y is prime, what is the units’ digit of 7^x + 9^y ?
(1) 24 < y < 32 (2) x = 1
 I get that just from the question stem itself, you know either x or y must be equal to 1, thus B cannot be the answer. I can't figure out how you can tell the units digit of 9 to the Y just from the statement one.
Help? My approach was: 1. Y can be either 29 or 31. Now y^29 or y^31 will always end in 9 and watever the power of X is the units digit is going to be the same. So A. 2. X=1 is not enough for a conclusion.
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Re: If x and y are positive integers such that the product of x [#permalink]
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17 Oct 2010, 04:23
The good thing is that 9^y when y is a prime number greater than 2, since all the primes greater than 2 are odd, the digit will always be 9.
So in the statement 1 it doesn't matter whether y is 29 or 31, we just need to know that y is not 2. Any range that doesn't contain 2 will be good.



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Re: If x and y are positive integers such that the product of x [#permalink]
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20 Oct 2010, 19:17
good one.... ans : A



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Re: If x and y are positive integers such that the product of x [#permalink]
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21 Aug 2017, 05:38
Atrain13gm wrote: If x and y are positive integers such that the product of x and y is prime, what is the units’ digit of 7^x + 9^y ?
(1) 24 < y < 32 (2) x = 1
 I get that just from the question stem itself, you know either x or y must be equal to 1, thus B cannot be the answer. I can't figure out how you can tell the units digit of 9 to the Y just from the statement one.
Help?
If x and y are positive integers such that the product of x and y is prime, what is the units’ digit of 7^x + 9^y?Since x and y are positive integers, then in order the product of x and y to be prime, either of them must be 1 another must be a prime number. (1) 24 < y < 32 > y is not equal to 1, thus y must be a prime number and x must be equal to 1. Only primes between 24 and 32 are 29 and 31, so y is either 29 or 31. Now, the units digit of 9^odd is 9, thus the units’ digit of 7^1 + 9^odd is 7+9=6. Sufficient. (2) x = 1 > y can be ANY prime number. If x=1 and y=2, then the units’ digit of 7^x + 9^y is 8, but if x=1 and y is any other prime then the the units’ digit of 7^x + 9^y is 6. Not sufficient. Answer: A. OPEN DISCUSSION OF THIS QUESTION IS HERE: http://gmatclub.com/forum/ifxandyar ... 42992.html
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